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Theorem finsumvtxdg2sstep 26676
Description: Induction step of finsumvtxdg2size 26677: In a finite pseudograph of finite size, the sum of the degrees of all vertices of the pseudograph is twice the size of the pseudograph if the sum of the degrees of all vertices of the subgraph of the pseudograph not containing one of the vertices is twice the size of the subgraph. (Contributed by AV, 19-Dec-2021.)
Hypotheses
Ref Expression
finsumvtxdg2sstep.v 𝑉 = (Vtx‘𝐺)
finsumvtxdg2sstep.e 𝐸 = (iEdg‘𝐺)
finsumvtxdg2sstep.k 𝐾 = (𝑉 ∖ {𝑁})
finsumvtxdg2sstep.i 𝐼 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}
finsumvtxdg2sstep.p 𝑃 = (𝐸𝐼)
finsumvtxdg2sstep.s 𝑆 = ⟨𝐾, 𝑃
Assertion
Ref Expression
finsumvtxdg2sstep (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((𝑃 ∈ Fin → Σ𝑣𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → Σ𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = (2 · (♯‘𝐸))))
Distinct variable groups:   𝑖,𝐸   𝑖,𝐺   𝑖,𝑁   𝑣,𝐸   𝑣,𝐺   𝑣,𝐾   𝑣,𝑁   𝑖,𝑉,𝑣
Allowed substitution hints:   𝑃(𝑣,𝑖)   𝑆(𝑣,𝑖)   𝐼(𝑣,𝑖)   𝐾(𝑖)

Proof of Theorem finsumvtxdg2sstep
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 finsumvtxdg2sstep.p . . 3 𝑃 = (𝐸𝐼)
2 finresfin 8353 . . . 4 (𝐸 ∈ Fin → (𝐸𝐼) ∈ Fin)
32ad2antll 767 . . 3 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (𝐸𝐼) ∈ Fin)
41, 3syl5eqel 2843 . 2 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑃 ∈ Fin)
5 difsnid 4486 . . . . . . . . 9 (𝑁𝑉 → ((𝑉 ∖ {𝑁}) ∪ {𝑁}) = 𝑉)
65ad2antlr 765 . . . . . . . 8 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((𝑉 ∖ {𝑁}) ∪ {𝑁}) = 𝑉)
76eqcomd 2766 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑉 = ((𝑉 ∖ {𝑁}) ∪ {𝑁}))
87sumeq1d 14650 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → Σ𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = Σ𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})((VtxDeg‘𝐺)‘𝑣))
9 diffi 8359 . . . . . . . . 9 (𝑉 ∈ Fin → (𝑉 ∖ {𝑁}) ∈ Fin)
109adantr 472 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → (𝑉 ∖ {𝑁}) ∈ Fin)
1110adantl 473 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (𝑉 ∖ {𝑁}) ∈ Fin)
12 simpr 479 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑁𝑉)
1312adantr 472 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑁𝑉)
14 neldifsn 4467 . . . . . . . . 9 ¬ 𝑁 ∈ (𝑉 ∖ {𝑁})
1514nelir 3038 . . . . . . . 8 𝑁 ∉ (𝑉 ∖ {𝑁})
1615a1i 11 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑁 ∉ (𝑉 ∖ {𝑁}))
17 dmfi 8411 . . . . . . . . . . 11 (𝐸 ∈ Fin → dom 𝐸 ∈ Fin)
1817ad2antll 767 . . . . . . . . . 10 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → dom 𝐸 ∈ Fin)
195eleq2d 2825 . . . . . . . . . . . . 13 (𝑁𝑉 → (𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁}) ↔ 𝑣𝑉))
2019biimpd 219 . . . . . . . . . . . 12 (𝑁𝑉 → (𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁}) → 𝑣𝑉))
2120ad2antlr 765 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁}) → 𝑣𝑉))
2221imp 444 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ 𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})) → 𝑣𝑉)
23 finsumvtxdg2sstep.v . . . . . . . . . . 11 𝑉 = (Vtx‘𝐺)
24 finsumvtxdg2sstep.e . . . . . . . . . . 11 𝐸 = (iEdg‘𝐺)
25 eqid 2760 . . . . . . . . . . 11 dom 𝐸 = dom 𝐸
2623, 24, 25vtxdgfisnn0 26602 . . . . . . . . . 10 ((dom 𝐸 ∈ Fin ∧ 𝑣𝑉) → ((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0)
2718, 22, 26syl2an2r 911 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ 𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})) → ((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0)
2827nn0zd 11692 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ 𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})) → ((VtxDeg‘𝐺)‘𝑣) ∈ ℤ)
2928ralrimiva 3104 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ∀𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})((VtxDeg‘𝐺)‘𝑣) ∈ ℤ)
30 fsumsplitsnun 14703 . . . . . . 7 (((𝑉 ∖ {𝑁}) ∈ Fin ∧ (𝑁𝑉𝑁 ∉ (𝑉 ∖ {𝑁})) ∧ ∀𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})((VtxDeg‘𝐺)‘𝑣) ∈ ℤ) → Σ𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + 𝑁 / 𝑣((VtxDeg‘𝐺)‘𝑣)))
3111, 13, 16, 29, 30syl121anc 1482 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → Σ𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + 𝑁 / 𝑣((VtxDeg‘𝐺)‘𝑣)))
32 fveq2 6353 . . . . . . . . . 10 (𝑣 = 𝑁 → ((VtxDeg‘𝐺)‘𝑣) = ((VtxDeg‘𝐺)‘𝑁))
3332adantl 473 . . . . . . . . 9 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑣 = 𝑁) → ((VtxDeg‘𝐺)‘𝑣) = ((VtxDeg‘𝐺)‘𝑁))
3412, 33csbied 3701 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑁 / 𝑣((VtxDeg‘𝐺)‘𝑣) = ((VtxDeg‘𝐺)‘𝑁))
3534adantr 472 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑁 / 𝑣((VtxDeg‘𝐺)‘𝑣) = ((VtxDeg‘𝐺)‘𝑁))
3635oveq2d 6830 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + 𝑁 / 𝑣((VtxDeg‘𝐺)‘𝑣)) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((VtxDeg‘𝐺)‘𝑁)))
378, 31, 363eqtrd 2798 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → Σ𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((VtxDeg‘𝐺)‘𝑁)))
3837adantr 472 . . . 4 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → Σ𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((VtxDeg‘𝐺)‘𝑁)))
39 finsumvtxdg2sstep.k . . . . . . . 8 𝐾 = (𝑉 ∖ {𝑁})
40 finsumvtxdg2sstep.i . . . . . . . 8 𝐼 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}
41 finsumvtxdg2sstep.s . . . . . . . 8 𝑆 = ⟨𝐾, 𝑃
42 fveq2 6353 . . . . . . . . . 10 (𝑗 = 𝑖 → (𝐸𝑗) = (𝐸𝑖))
4342eleq2d 2825 . . . . . . . . 9 (𝑗 = 𝑖 → (𝑁 ∈ (𝐸𝑗) ↔ 𝑁 ∈ (𝐸𝑖)))
4443cbvrabv 3339 . . . . . . . 8 {𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)} = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}
4523, 24, 39, 40, 1, 41, 44finsumvtxdg2ssteplem2 26673 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((VtxDeg‘𝐺)‘𝑁) = ((♯‘{𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}})))
4645oveq2d 6830 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((VtxDeg‘𝐺)‘𝑁)) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((♯‘{𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}}))))
4746adantr 472 . . . . 5 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((VtxDeg‘𝐺)‘𝑁)) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((♯‘{𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}}))))
4823, 24, 39, 40, 1, 41, 44finsumvtxdg2ssteplem4 26675 . . . . 5 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((♯‘{𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}}))) = (2 · ((♯‘𝑃) + (♯‘{𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)}))))
4944fveq2i 6356 . . . . . . . 8 (♯‘{𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)}) = (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)})
5049oveq2i 6825 . . . . . . 7 ((♯‘𝑃) + (♯‘{𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)})) = ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}))
5150oveq2i 6825 . . . . . 6 (2 · ((♯‘𝑃) + (♯‘{𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)}))) = (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)})))
5251a1i 11 . . . . 5 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (2 · ((♯‘𝑃) + (♯‘{𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)}))) = (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}))))
5347, 48, 523eqtrd 2798 . . . 4 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((VtxDeg‘𝐺)‘𝑁)) = (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}))))
54 eqid 2760 . . . . . . . 8 {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)} = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}
5523, 24, 39, 40, 1, 41, 54finsumvtxdg2ssteplem1 26672 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (♯‘𝐸) = ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)})))
5655oveq2d 6830 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (2 · (♯‘𝐸)) = (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}))))
5756eqcomd 2766 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}))) = (2 · (♯‘𝐸)))
5857adantr 472 . . . 4 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}))) = (2 · (♯‘𝐸)))
5938, 53, 583eqtrd 2798 . . 3 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → Σ𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = (2 · (♯‘𝐸)))
6059ex 449 . 2 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (Σ𝑣𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃)) → Σ𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = (2 · (♯‘𝐸))))
614, 60embantd 59 1 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((𝑃 ∈ Fin → Σ𝑣𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → Σ𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = (2 · (♯‘𝐸))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wnel 3035  wral 3050  {crab 3054  csb 3674  cdif 3712  cun 3713  {csn 4321  cop 4327  dom cdm 5266  cres 5268  cfv 6049  (class class class)co 6814  Fincfn 8123   + caddc 10151   · cmul 10153  2c2 11282  0cn0 11504  cz 11589  chash 13331  Σcsu 14635  Vtxcvtx 26094  iEdgciedg 26095  UPGraphcupgr 26195  VtxDegcvtxdg 26592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-disj 4773  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-2o 7731  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-sup 8515  df-oi 8582  df-card 8975  df-cda 9202  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-nn 11233  df-2 11291  df-3 11292  df-n0 11505  df-xnn0 11576  df-z 11590  df-uz 11900  df-rp 12046  df-xadd 12160  df-fz 12540  df-fzo 12680  df-seq 13016  df-exp 13075  df-hash 13332  df-cj 14058  df-re 14059  df-im 14060  df-sqrt 14194  df-abs 14195  df-clim 14438  df-sum 14636  df-vtx 26096  df-iedg 26097  df-edg 26160  df-uhgr 26173  df-upgr 26197  df-vtxdg 26593
This theorem is referenced by:  finsumvtxdg2size  26677
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